Extra-Stage Cube Network Reliability Estimation Using Stratified Sampling Monte Carlo Method

Transcription

1 nlne a h://ejm.fskm.m.ed.my Vol., No March -8 Engneerng e-transacon, Unversy of Malaya Exra-Sage Cbe Nework Relably Esmaon Usng Srafed Samlng Mone Carlo Mehod ndra Gnawan, Sellaan Palanaan, Lm Choo Sen Malaysa Unversy of Scence & Technology Emal: {gnawan, sell, cslm}@ms.ed.my ABSTRACT The aer resens he Exra-Sage Cbe nerconnecon nework relably esmaon sng Srafed Samlng Mone Carlo (SSMC) mehod. The arameer of relably ncldes ermnal relably, broadcas relably and nework relably. The SSMC mehod rovdes aroxmae solons by erformng sascal samlng exermens on a comer. Ths mehod of relably redcon s sefl when sysem comlexy makes he formlaon of exac models essenally mossble. A comer rogram has been develoed and he SSMC mehod was demonsraed as a good esmaor for all he hree yes of he relably arameer afer comared wh he exac relably and relably bonds. Keywords: Srafed Samlng Mone Carlo, Mlsage nerconnecon Nework, Exra-Sage Cbe Nework, Termnal Relably, Broadcas Relably, Nework Relably. nrodcon Hgh comng ower s needed o solve he roblems n several areas sch as defense, aerosace, ransoraon and elecommncaons. However, he execon rae reqred s beyond he caables of crren large comer sysem. Therefore, he sysem erformance can only sgnfcanly mroved hrogh addonal concrren rocessng, whch s arallel comers wh mlle rocessors ha can sly he sor essenal o acheve he comaonal erformance goals for hese alcaons. A arallel mlrocessor sysem lays he mechansm needed for ransferrng nformaon beween rocessor and memory modles, namely he nerconnecon nework. nerconnecon neworks offer an aracve and economcal solon o he commncaon and nerconnecon beween sysem comonens. Generally can be caegorzed no for major classes based on nework oology []: Shared-medm neworks, drec neworks, ndrec neworks and hybrd neworks. Mlsage nerconnecon Neworks (MN) s one of he sbclasses of he ndrec nework. MN s sed n mlrocessor sysems o nerconnec rocessors and memory modles. n he as decade, Mlsage nerconnecon Neworks (MNs) was aled n felds sch as elecommncaon and arallel comng []. A MN connecs N ns (sorces) o N os (desnaons) hrogh a nmber of swch sages and s referred o as an N x N MN. The arameer N s called he sze of he nework. There are nmeros dfferen MN, ncldng Dela nework [], Bnary n-cbe nework [], Mlsage Cbe Nework, Exra-Sage Cbe nework [], Shffle-Exchange nework, mega nework [], Gamma nework [] and ec. Ths aer wll focs on Exra-Sage Cbe (ESC) nework and esmae he relably of ESC nework sng Srafed Samlng Mone Carlo (SSMC) mehod. Exra-Sage Cbe Nework Relably The Exra-Sage Cbe (ESC) nework [] s formed from he generalzed cbe [] by addng an exra sage o he nework along wh mllexers and demllexers a he n and o sages, resecvely, as shown n Fg.. ESC s a -Pah MN ha rovdes nqe ahs from he n sde o he o sde of he MN. The redndan ahs fncon as alernae ranson ah from any n o any o, when any swchng elemen fals. Generally, he nmber of sages, n s log N +.

2 nlne a h://ejm.fskm.m.ed.my Vol., No March -8 Engneerng e-transacon, Unversy of Malaya Sage n s conneced lke sage, where lnks ha vary n he low-order b are ared. For examle, ( n bnary b) s ared wh ( n bnary b) where he low-order b (b wh alc fon) s dfferng. n he followng secons, he ESC nework wll be shown who he mllexers o smlfy he drawng. n Sage Fg. The Exra-Sage Cbe Nework for N = 8 There are manly hree yes of relably measres, whch are moran o MNs, namely ermnal relably, broadcas relably and nework relably. Termnal Relably, generally sed as a measre of robsness of a MN, s he robably of exsence of a leas one fal free ah beween a desgnaed ar of n (s) and o () ermnals [9]. Broadcas relably s he robably of a MN n broadcasng daa from a gven n ermnal o all of he o ermnals of he nework. A nework s sad o have faled when a connecon canno be made from a gven n ermnal o a leas one of s o ermnals [9]. Nework relably (all ermnal relably) s he robably ha here exss a connecon beween each n o all os. The nework s consdered o be oeraonal as long as every sorce can commncae wh each desnaon [9]. A nmber of assmons are made o evalae he relably of ESC:. All falres are sascally ndeenden.. All swchng elemens (SEs) are sbsanally less relable han he lnks. Even hogh he lnks can fal, hey are generally mch more relable han he SEs.. Each SE n he MN has only wo ossble saes, workng or faled. Sose s defned: = SE workng, = SE faled.. The relably of he SE beng n a arclar sae s known.. All SEs are dencal and have consan exonenal falre raes.. The SEs canno be reared.. All SEs a sage and n ms work for he sysem o be oeraonal. 8. A SE s assmed as faled when canno erform any of he connecon fncons (sragh, exchange/swa, erbroadcas, and lower-broadcas). Srafed Samlng Mone Carlo Algorhm. Srafed Samlng Technqe Srafed Samlng echnqes wll be aled o ncrease effcency of he Mone Carlo mehod. The conce of Mone Carlo exermen had been broadened o he general roblem of evalang he Lebesqe-Seljes negral [] ζ = K(z)dF(z) () R where z = (z,, z s ), {F(z)} s a jon dsrbon fncon on he s-dmensonal regon R and {K(z)} denoes a weghng kernel defned on R. Nex, le R,, R h denoe dsjon sbses of R n exresson () sch ha R = U h R = and consder ζ here wren n he alernave, b eqvalen form ζ = ζ = = h = R R ζ K(z)dF()(z) df(z) df () (z) = - df(z) R h and + + h = Hereafer, we refer o R as sram. Sose ha one generaes m ndeenden samles Z (,), ( m, Z, ) from sram for each =,, h. Then h m ζ (m + + m h ) = = m gves an nbased esmaor of ζ wh var ζ (m + + m h ) = h = j = K(Z (j,) ) σ / m

3 nlne a h://ejm.fskm.m.ed.my Vol., No March -8 Engneerng e-transacon, Unversy of Malaya where σ = E[K(Z (j,) ) - ζ ] = R [K(z) - ζ ] df(z) h Srafed samlng ncldes any samlng lan ha arons R n hs way and fxes he nmber of samles o be aken n each sram. Presmably, hs aron and samle sze selecon lead o var ζ (m + + m h ) < var ζ m where m = (m + + m h ). For fxed m, he nmber of samles m o be generaed n sram, h s deermned by m = m h and var ζ (m + + m h ) var where = k (-) k- ζ m. Relably Esmaon Algorhm Mone Carlo Algorhm wh Srafed Samlng Technqe (Algo ) n: SE relably (r), Nmber of sages (n), Nmber of rals/samle sze (m), Nmber of SE for nermedae sages (f) Nmber of SE for frs and las sages (f) : Relably (R) Algorhm: SET f = Nmber of nermedae sage SE SET _ [] = CALL Algo (f, r) SET m_ [] = array wh _.lengh elemen SET workng_connecon = SET condced_es = NT neworkarray[n][ ] FR = o _.Lengh- m_[] = Mah.Floor[(_).(m)] F m_[] = CNTNUE ELSE SET [] = array wh f elemen FR z= o - [z] = FR j = o m_[]- SET aorand = random objec wh a mergeneraed seed FR = o fl- SET v = + Mah.Floor[(aoRand.NexDoble).(f-)] SET w = [] [] = [v] [v] = w SET = SET Frs AND Las sages SE = FR g = o neworkarray.lengh- FR h = o neworkarray[g].lengh- neworkarray[g][h] = N[] NCREMENT F nework s oeraonal NCREMENT workng_connecon END F END F condced_es = condced_es + m_[] SET nermedae_r=workng_connecon/condced_e s R = (r f ).(nermedae_r) Generae Probably Array (Algo ) n: SE relably (r), Nmber of swches for nermedae sages (nm) : Probably Array (_[]) Algorhm: Algo. Ge _ (nm, r) SET _[] = array wh nm+ elemen FR = o nm SET a = r SET b = (-r) nm- _[] = [CALL Algo. (nm,)].a.b RETURN _ Algo. Ge Bnomal coeffcen (nm, k) SET b[] = array wh nm+ elemen SET b[] = FR = o nm SET b[] = FR j = - o j> b[j] = b[j] + b[j-] j = j- RETURN b[k] The algorhms above are wren sng C# synax. The esmaon for ermnal relably, broadcas relably and nework relably sng SSMC mehod s smlar n erms of general algorhm, exce he confgraon of each ye of relably (See Fg., Fg. & Fg.), whch reles

4 nlne a h://ejm.fskm.m.ed.my Vol., No March -8 Engneerng e-transacon, Unversy of Malaya on he relably defnon. Therefore, he nmber of frs and las sages, and nermedae sages s dfferen accordng o he confgraon. n Sage Fg. nalzed neworkarray Reresenng Termnal Confgraon for 8 x 8 ESC Nework n Nmercal Resls The resls obaned for he ermnal relably (R ST ) sng SSMC mehod wh,, rals were comared agans exac ermnal relably obaned by Mahemacal Formla (MF) []. The measre of accracy sng arameer α s defned as α = SSMC MF MF Fg. shows ha he resls for N= ha obaned sng SSMS mehod s very close o he exac ermnal relably. Lower α ndcaes hgher accracy. From he resls for N=8 o N=8, α obaned are all less han.. As he SE relably r ncreasng, arameer α s decreasng. Therefore SSMC s a good esmaon mehod for ermnal relably of small sze o large sze ESC nework Sage Termnal Relably Fg. nalzed neworkarray Reresenng Broadcas Confgraon for 8 x 8 ESC Nework Swchng Elemen Relably (r) Srafed Samlng Mone Carlo Mahemacal Formla N P U T Sage Fg. nalzed neworkarray Reresenng Nework Confgraon for 8 x 8 ESC Nework The nework oeraonal checkng s anoher algorhm ha wll no dscss n deal n hs aer. Take noe ha he sages label for he esmaon algorhm s he reverse of he ESC nework defnon, so ha he algorhm s easer o handle. U T P U T Fg. Termnal Relably (R ST ) for ESC Nework, N= Table Termnal Relably Resls for N = SE Relably (r) R ST SSMC (,, rals) Mahemacal Formla Measre of Accracy (α) = (SSMC - MF) / MF The resls obaned for he broadcas relably (R B ) sng SSMC mehod wh, rals were comared agans exac broadcas relably

5 nlne a h://ejm.fskm.m.ed.my Vol., No March -8 Engneerng e-transacon, Unversy of Malaya obaned by Marx Enmeraon (ME) mehod [] for N=8 o N=, and by Recrsve Manner (RM) mehod [] for N=8 o N=. The measre of accracy sng arameer α s defned as α = SSMC ME for N=8 o N= ME α = SSMC RM for N=8 o N= RM Fg. shows ha he resls for N= ha obaned sng SSMS mehod s very close o he exac broadcas relably and falls beween lower bond and er bond [,8]. Lower α ndcaes hgher accracy. α obaned are all less han.8 and decreases when r ncreases. These sasfacon resls show ha he SSMC s a good esmaon mehod for broadcas relably of small sze o large sze ESC nework f hey fall beween he lower bond and er bond []. The measre of accracy sng arameer α s defned as α = SSMC ME for N=8 o N= ME Fg. shows ha he resls for N= ha obaned sng SSMS mehod s falls beween lower bond and er bond of nework relably. The esmaon has larger varance when he N ncreases Swchng Elemen Relably (r) Srafed Samlng M one Carlo Lower Bond (Blake & Trved M ehod) Uer Bond (Blake & Trved M ehod) New Lower Bond (Cheng & be M ehod) New Uer Bond (Cheng & be M ehod) Swchng Elemen Relably (r) Srafed Samlng M one Carlo Recrsve M anner Lower Bond Uer Bond Fg.Broadcas Relably (R B ) for ESC Nework, N= Table Broadcas Relably Resls for N = SE Relably (r) SSMC (, rals) R B Recrsve manner(cheng and be) Measre of Accracy (α) = (SSMC - RM) / RM The resls obaned for he nework relably (R W ) sng SSMC mehod wh, rals were comared agans exac broadcas relably obaned by Marx Enmeraon (ME) mehod [] for N=8 and N=. For larger sze ESC nework, R W obaned s consdered accrae and acceable Fg.Nework Relably (R W ) for ESC Nework, N= Table Nework Relably Resls for N = R W SE Relably (r) SSMC (, rals) Lower Bond (Blake & Trved) Uer Bond (Blake & Trved) Conclson The comarson and analyss showed ha he Srafed Samlng Mone Carlo mehod rovdes sasfed resls for ermnal relably, broadcas relably and nework relably of small sze o large sze ESC nework.

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